 Hi and welcome to the session, I am Arsha and I am going to help you solve this problem of your book which says, check whether 7 plus 3x is a factor of 3x cube plus 7x. So before solving this problem, let us first learn what does remainder theorem say. With the help of this theorem, we will check whether 7 plus 3x is a factor of 3x cube plus 7x or not. So this theorem says if px is any polynomial of degree greater than or equal to 1, then when px is divided by x minus a, where a is any real number, then remainder is equal to p at a. So this theorem is a clear idea that we will be using in this problem to solve it. So let us now start with the solution and let us denote 3x cube plus 7x by px. Here we have to check whether 7 plus 3x which is the linear polynomial, a factor of px. So first we will find the 0 of 7 plus 3x, 0 of 7 plus 3x is minus 7 upon 3, since to find the 0 of any linear polynomial, we equate it to 0 and upon doing so we get x is equal to minus 7 upon 3. Now by remainder theorem, 7 plus 3x is a factor of px if p at minus 7 upon 3 is equal to 0. So what we will do is, we find the value of the polynomial px at minus 7 upon 3. So we have 3 times of minus 7 upon 3 whole cube plus 7 into minus 7 upon 3, which is further equal to 3 minus 7 whole cube is minus 343 and 3 whole cube is 27 plus 7 into minus 7 is minus 49 and the denominator we have 3. 3 is the common factor in the numerator and denominator, so on cancelling we have 3 1s 3 9s are 27, therefore minus 343 upon 9 plus minus 49 upon 3 which is further equal to minus 490 upon 9, since on taking LCM of 9 and 3 we have 9 and when 9 is divided by 9 we get 1, so multiplying the numerator minus 343 by 1 gives minus 343, then we have plus 9 and on dividing 9 by 3 we get 3, therefore multiplying the numerator minus 79 by 3 and hence on solving we get minus 490 upon 9 which is not equal to 0. So this implies that p at minus 7 upon 3 is not equal to 0. This shows that 7 plus 3x is not a factor of px, as the remainder is not 0, 7 plus 3x is not a factor of x cube plus 7x, no since the remainder is not 0 is our answer. This completes the solution, remember the remainder theorem while you doing these types of problem and also be careful with the calculations, so hope you enjoyed this session, take care and have a good day.